Exercise · ZW-LAW-OF-TOTAL-PROBABILITY-010 · ★★★ Advanced
A disease affects 2 % of a population. The test is positive for 90 % of the sick and 5 % of the healthy. What fraction of everyone tests positive?
Choose one option.
Show answer
- C. 0.067
Explanation
\(0.02\cdot0.90+0.98\cdot0.05=0.018+0.049=0.067\). False positives among the healthy outweigh the true positives.
Review the concept: Law of total probability
The total probability of an event \(A\) is the weighted average of its conditional probabilities \(P(A\mid B_i)\) over a partition \(B_i\) of the sample space, with the probability of each case as its weight.
\[P(A)=\sum_{i=1}^{n}P(A\mid B_i)\,P(B_i)\]
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